Sphere-dome construction

ABSTRACT

A self-supporting dome-like or sphere-like structure made of interconnected linear or planar elements, comparable to a geodesic dome and comprising, in the full sphere approximation, 12 regular pentagons and, in the simplest case, 60 irregular but identical hexagons. For higher orders of the structure, the number of pentagons remains 12 but the number of hexagons increases according to the formula

United States Patent Langner [451 Oct. 10,1972

[54] SPHERE-DOME CONSTRUCTION [72] Inventor: Horst Egon Phillip Langner,Calgary, Alberta, Canada [73] Assignee: Langner Domes Ltd., Regina,

Saskatchewan, Canada [22] Filed: Sept. 28, 1970 [21] Appl. No.: 76,107

[52] US. Cl ..52/80, 52/DIG. 10 [51] Int. Cl. ..E04b U342 [58] Field ofSearch ..52/80, 81, DIG. 10; 35/46 A, 35/47 [56] References Cited UNITEDSTATES PATENTS 3,304,669 2/1967 Geschwender ..52/DIG. 10 3,043,0547/1962 Schmidt ..52/81 2,918,992 /1959 Gelsavage ..52/81 185,889 1/1877Boorman ..35/46 A 3,197,927 8/1965 Fuller ..52/20 3,359,694 12/1967l-Iein ..52/81 OTHER PUBLICATIONS Cosmo-Hut May 19, 1965, 4 pages, CosmoManufacturing 20201 Hoover Rd., Detroit 5, Michigan.

Mathematical Models Cundy and Rollett QA 11c 8 1961 p. 110- 111 secondedition.

Primary Examiner-Frank L. Abbott Assistant Examiner-Leslie A. BraunAttorney-Smart & Biggar [57] ABSTRACT A self-supporting dome-like orsphere-like structure made of interconnected linear or planar elements,comparable to a geodesic dome and comprising, in the full sphereapproximation, 12 regular pentagons and, in the simplest case, 60irregular but identical hexagons. For higher orders of the structure,the number of pentagons remains 12 but the number of hexagons increasesaccording to the formula where M, is the number of hexagons in a sphereapproximation of order k, and N is the number of hexagons in a sphereapproximation of order (k-l 26 Claims, 7 Drawing Figures PATENTEDHN 10 92 3.696. 566

SHEET 1 OF 4 VENTOR HORST E. ANGNER Bg jr ATTORNEYS.

' PATENTED OCT 10 I97? 3, 6 96, 5 6 6 NNNNNN OR HORST E. P NNNNN ERTTTTTTT YS PATENTEDncI 0 I972 SHEET 3 UF 4 FIG? INVENTOR HORST E. PLANGNER MV'W ATTORNEYS.

PATENTEDUCT 10 I972 3, 696, 566

sum 14 0F 4 INVENTOR HORST E. P LANGNER ATTORNEYS SPHERE-DOMECONSTRUCTION BACKGROUND OF THE INVENTION This invention relates toself-supporting building structures and particularly to structuresapproximating the shape of the sphere or a portion of a sphere (such asa dome), of the type composed of interconnected linear or planarelements.

A number of building structures are known in which a sphere or portionof a sphere is approximated by a plurality of interconnected linear ribsor interconnected plane panels. The structures have in common withstructures of the present invention the fact that no internal supportingelements are usedthe outer shell of interconnected panels or ribs issufficient to support the structure.

The most famous of this class of building structure is the geodesic domeof Buckminster Fuller, the basic concept of which is described in hisU.S. Pat. No. 2,682,235, issued June 29, I954, and variants of whichhave been described in later patents and other literature. Among theseis Fuller's U.S. Pat. No. 3,197,927 (Aug. 3, I965) which teaches the useof hexagonalpentagonal structural arrangements. Other literature alsodescribes hexagonal-pentagonal sphere-approximating structures, forexample U.S. Pat. No. 2,918,992 (.I. Z. Gelsavage, Dec. 29, I959). Theseknown structures can all be classified as geodesic structures, i.e., ifribs are used, the ribs tend to align themselves along great circles ofthe sphere which is approximated by the structure, and if planarelements are used, the great circles are approximated by the side edgesof the individual planar elements (or, by the lines joining the centersof the planar elements).

One of the disadvantages associated with such previously knownstructures is that the geodesic alignment of the structural elementscreates fold lines about which the entire structure can collapse whensubjected to sufficiently severe stresses. Regularity in the structuraldesign is advantageous from the point of view of enabling the use of asmall number of different modules, i.e., different sizes and shapes ofcomponent elements, for the composition of the structure, but thisstructural regularity can be disadvantageous, from the point of view ofoverall strength of the structure, if the regularity produces structuralweaknesses such as fold lines along great circles of the sphere whichthe structure approximates.

Other proposals such as that contained in D. G. Emmerichs U.S. Pat. No.3,341,989 (Sept. 19, 1967) have involved the use of quadrangles,hexagons, pentagons or more complex elements but have either failed toovercome the structural strength problem just men tioned or have failedto create a pleasing aesthetic effect. One of the contributing factorsto aesthetic appeal is considered by the inventor to reside in the useof elements of approximately the same size. Construction of a buildingis usually facilitated as well when the component elements are ofapproximately the same size, and when the number of different sizes andshapes needed is small.

SUMMARY OF THE INVENTION It is an object of the present invention tocombine in an aesthetically attractive sphere-approximating structure,or a portion thereof, a plurality of pentagons and hexagons formed byinterconnected planar or linear elements in a sphere-approximatingconfiguration, wherein all of the'component elements are ofapproximately the same size, wherein a small number of different sizesand shapes of elements are required, and wherein structural strength isachieved by the choice of the hexagonal or pentagonal shape ofindividual elements and also by the configurationof these elements intheir sphere-approximating structural relationship.

In its broadest aspect, the invention provides the whole or a portion (aportion being, for example, a dome) of a self-supportingsphere-approximating structure comprising, in its completely enclosedsphere-approximating whole version, 12 equal regular pentagons havingtheir centers at the vertexes of a regular icosahedron; and 60 (for asphere approximation of order 1) or M, (for a sphere approximation oforder k hexagons, where k is an integer greater than I, and, asindicated, M, is the number of hexagons in a whole sphere-approximatingstructure of order k. At least some of the hexagons depart slightly fromregular hexagons to enable continuity of the structure.

Generally, sphere-approximating structures of the first, second andthird orders will be found to be satisfactory for most structuralpurposes. As can be seen from the above relationship, asphere-approximating structure of the second order would have twohundred hexagons, and of the third order would have six hundred andtwenty hexagons.

SUMMARY OF THE DRAWINGS FIG. 1 shows a sphere-approximating structure ofthe first order, in accordance with the present invention.

FIG. 2 shows a sphere-approximatin g structure of the second order,according to the invention.

FIG. 3 is a diagram illustrating the triangulation technique accordingto which sphereapproximating structures of higher order can beconstructed from sphere-approximating structures of lower orderaccording to the invention.

FIG. 4 illustrates the two component planar elements of asphere-approximating structure of the first order, according to theinvention.

FIG. 5 illustrates the component elements of a sphere-approximatingstructure of the second order according to the invention.

FIG. 6 illustrates the planar component elements of asphere-approximating structure of the third order according to theinvention.

FIG. 7 (on the same sheet as FIG. 3) illustrates in side elevation astructure including a portion of a sphere-approximating structure of thefirst order constructed in accordance with the present invention.

DETAILED DESCRIPTION WITH REFERENCE TO THE DRAWINGS FIG. 1 illustrates asphere-approximating structure of the first order constructed inaccordance with the invention. The complete sphere approximationincludes 12 pentagons and 60 hexagons. The 12 pentagons, each of whichshown in FIG. 1 is designated as P11, are re gular pentagons placed onthe sphere-approximating structure so that the centers of the .pentagonsare located at the vertexes of a regular icosahedron. The icosahedronand corresponding invert dodecahedron interrelationship with the firstorder structure as embodied in FIG. 1 can best be understood byobserving that pentagon centers V1, V2, V3, V4, V5, V6 etc. definevertexes of a regular icosahedron, while points V7, V8, V9, V10, V11 arelocated at the vertexes of a regular dodecahedron which is the invert ofsuch regular icosahedron.

In FIG. 1 each pentagon is surrounded by five contiguous irregularhexagons each designated as H11. The result is that, since there arefive hexagons for each pentagon, there are 60 hexagons in the completesphere-approximating structure, each of which is exactly the same sizeand shape, but in order that all elements be contiguous, the hexagonsmust part slightly from regular hexagons. However, they do not depart somarkedly from regular hexagons that the overall aesthetic result of thestructure of FIG. 1 is unpleasant.

In order that the relative size and shapes of the component elements ofFIG. 1 be contiguous, so as to form a completely enclosed structureapproximating a sphere, the interior angles and the lengths of the sidesof hexagons H11 must be carefully selected. One approach that can betaken is to consider each of the pentagons P11 and hexagons H11 ascircumscribing a circle (as for example the inscribed circle C in FIG.1), and to consider that the circles, which will be tangent to oneanother in this hypothesis, are in turn tangent to the surface of acommon sphere. On this assumption, it can be established that the circleinscribed in each of the pentagons P11 can be considered to be thecircular face of a right regular cone whose apex angle is about 950 Theinscribed circle in each of the hexagons can be considered to be thecircular planar face of a right regular cone whose apex angle is about1324Vz with the circular radius being about 1.37 times that of theinscribed circle in the pentagon. It can be shown that cones havingthese dimensions can be stacked so that their circular faces are in factsubstantially tangent to one another and are substantially tangent to acommon sphere which the entire structure approximates.

This is not to suggest that other arrangements of cones or contiguouscircles could not be made to approximate a sphere nor should it beinferred that other arrangements of pentagons and hexagons could not bemade to form a sphere-approximating structure. However, the specificarrangement herein described does have the advantage of requiring onlytwo different sizes of planar components (the cone arrangement requiresonly two different sizes of cones or of contiguous circles) which hasthe advantage of simplicity for the purposes of mass production ofcomponents and also from the point of view of simplicity of erection ofthe structure by workmen.

FIG. 2 illustrates a sphere-approximating structure of the second orderaccording to the invention. Again agonal modules are designatedrespectively as H21, H22, and H23 in FIG. 2.

It will be noted that the side edges of the hexagonal and pentagonalelements of FIGS. 1 and 2 are not aligned along great circles orapproximate great circles. This is also true of the line joining centersof adjacent elements-the line may tend to follow a great circle for afew elements (see line L, for example, in FIG. 2) but a departure fromthe great circle eventually occurs.

In order to arrive at the FIG. 2 arrangement, the process oftriangulation may be applied to the configuration of FIG. 1. This isillustrated in FIG. 3. FIG. 3 is a diagram which divides thesphere-approximating surface shown into three sectors designated X, Y,and Z respectively. In the sector X, the configuration is that of FIG.1, namely a sphere-approximating structure of the first order. In sectorY, the first order configuration is shown in broken lines. From thisstructure, by conventional triangulation, one can derive a breakdown ofthe structural units into smaller structural units, again comprisingonly pentagons and hexagons. These are shown in solid lines in sector Yof FIG. 3. Finally, in sector 2 of FIG. 3, the configuration of order 2is shown in broken lines and, by conventional triangulation, the solidline configuration shown in sector Z of FIG. 3 represents asphere-approximating structure of the third order, according to theinvention.

Adjustments may have to be made in the sizes and shapes of the hexagonsproceeding by triangulation from a configuration of one order to theconfiguration of the next highest order, in order to bring about propercontiguity of adjacent elements in a three-dimensional array. However,the adjustments required going merely from one order to the next highestorder are relatively minor.

It can be shown that the number of hexagons for sphere-approximatingstructures of any given order according to the invention is twenty morethan three times the number of hexagons in the next lowest order ofsphere-approximating structure according to the invention, thus:

where N is the number of hexagons for a sphere-approximating structureaccording to the invention of order k, and k is an integer greaterthan 1. As explained above, the number of hexagons in the first orderstructure is 60. The above relationship implies that the second ordersphere approximating structure will have 200 hexagons and the thirdorder structure will have 620 hexagons. Regardless of the order, thenumber of pentagons is always l2.

FIGS. 4, 5, and 6 show examples of the planar modules than can be usedin sphere-approximating structures according to the invention of thefirst, second, and third order respectively. The pentagonal modules P11(first order--FIG. 4) P21 (second order-FIG. 5) and P31 (thirdorder-FIG. 6) are all regular pentagons. For each of the hexagonalmodules, sides a, b, c, d, e, andf and interior angles a, [3, 'y, 8, e,and can have the following relationships to one another, in accordancewith exemplary embodiments of the present invention:

Relative Side Lengths a b c d e f First Order H11 248 263 278 278 278263 Second Order H21 127 134 150 159 150 134 H22 159 159 159 159 159 159H23 150 150 159 159 159 159 Third Order H31 64.8 67.9 77.8 84.5 77.867.9 H32 84.5 84.5 87.3 90.1 87.3 84.5 H33 84.5 87.3 90.7 91.2 92.1 88.2H34 91.2 90.3 91.2 92.1 92.1 92.1 H35 90.1 90.7 90.3 90.3 90.3 90.7 H3677.8 77.8 84.5 84.5 84.5 84.5 H37 92.1 92.1 92.1 92.1 92.1 92.1

Angles a [3 7 8 e First Order H11 12344 12344' 11808' 11808' 11808'11808' Second Order H21 12523 12523' 12019' 11418 11418' 12019' H22 120120 120 120 120 120 H23 12351 12344' 12351 11808' 11808' 11808' ThirdOrder H31 12551' 12551 12128' 11241 l1241'l2128' H32 12105' 12105' 1210511750' 11750' 12105' H33 12105 12105' 12329' 11859 11951 11951' H3411951'1203812038 11951' 11951' 11951' H35 11805' 11805' 12357 11808'11808 l2357' H36 12553 11640' 12553' 11718' ll718' 11739' H37 120 I20120 120 120 120' Since a regular hexagon has interior angles of 120, itcan be seen from the above that the hexagons used in the sphereapproximation according to the invention depart only slightly fromregularity. The near approximation of regular hexagons is believed bythe inventor to contribute to the aesthetic appeal of the structures.

When the component planar elements have been assembled to form a sphereor portion of a sphere, it has been found, referring to FIG. 1, that theangle between the line joining the center (e.g., VI) of a pentagon toone of the vertexes (e.g., V in FIG. 1) of the corresponding invertdodecahedron and the line joining the center V1 to the pentagon vertex(V12) nearest V10, designated 6 in FIG. 1, is about 136' for theexamples herein discussed. This appears to be a constant angle at leastfor the first, second, and third orders.

In a building structure, the pentagons and hexagons can be formed byplanar sheets of the required shape, or the structure may be composed ofribs following the side edges of the pentagons and hexagons illustratedin the drawings. (Obviously, curved rather than planar sheets could beused so as to simulate a sphere more exactly). It is useful to considerthe geometric properties of the structure according to the invention onthe assumption that planar sheets are used but it will be understoodthat structures can be built according to the invention without usingplanar elements.

FIG. 7 illustrates an exemplary use of the principles of the presentinvention in a composite building structure. It can be seen that thedesign postulates a lower rectangular structure 71, an elongated terrace73, and a dome 75 constructed in accordance with the principles of thepresent invention. The dome takes the form of a segment of asphere-approximating structure according to the invention. Thepercentage of an entire sphere-approximation chosen by a designer forany given structural use is largely a matter of desired volume or areaand aesthetics. It is contemplated in accordance with the invention thatthere may be openings in or additions to the basic structure-forexample, FIG. 7 illustrates two different types of design modifications.The first type of modification involves a breaking of the structurealong the side edges of component portions of the structure to form awindow 77. The

designer may additionally choose to modify the structure by departingcompletely from the side edges of the structure and adding geometry ofquite a different kind-for example, the rectangular porch 79.

The structure according to the invention may be used as a tank or forother bulk storage, or for housing, arenas or other commercialbuildings. The intended uses and the materials suitable for constructionare analogous to those appropriate to geodesic domes and spheres.

What I claim as my invention is:

l. A self-supporting sphere-approximating structure comprising,interconnected structural elements forming twelve equal regularpentagons having their centers located at the vertexes of a regularicosahedron, and, for a sphere approximation of order 1, 60 hexagons;and for a sphere approximation of order k, N k hexagons at least some ofwhich depart from regular hexagons to enable contiguity of adjacent onesof said structural elements, where k is an integer greater than 1, whereN equals 3N,,., 20, and where N is the number of hexagons for a sphereapproximation of order one less than k.

2. A self-supporting sphere-approximating structure, comprisinginterconnected structural elements forming twelve equal regularpentagons and n hexagons at least some of which hexagons are irregularto enable contiguity of adjacent modules, the centers of the pentagonsdefining vertexes of a regular icosahedron, where n is an integerselected from the following group: 60, 200, and 620.

3. A structure as defined in claim 2 wherein the structural elements areplanar pentagonal and hexagonal modules. 2

4. A structure as defined in claim 2 wherein the structural elements areribs which lie along the edges of the hexagons and pentagons.

5. A structure as defined in claim 2, wherein n 60 and all of thehexagons are of the same size and the same shape.

6. A structure as defined in claim 2, wherein n 200 and hexagons areregular and of equal size, while 60 hexagons are of a firstpredetermined size and shape, and 60 hexagons are of a secondpredetermined size and shape.

7. A structure as defined in claim 5 wherein a circle may be inscribedin each hexagon and in each pentagon, and the structural elements arealigned such that circles inscribed in the hexagons and pentagons wouldbe tangent to a common sphere.

8. A structure as defined in claim 6 wherein the structural elements arealigned such that closed curved figures, selected from the class offigures comprising circles and approximate circles, inscribed in thehexagons and pentagons, would be approximately tangent to a commonsphere.

9. A structure as defined in claim 6 wherein the arrangement of thestructural elements is obtained substantially by triangulation of thatstructure obtained where the following conditions are satisfied: n isselected to be 60 and all of the hexagons are of the same size and sameshape.

10. A structure as defined in claim 2 wherein the angle between the linejoining the center of any of said pentagons to a vertex of adodecahedron corresponding in invert relationship to said icosahedronand the line joining the center of the last mentioned pentagon to thevertex of the pentagon nearest said dodecahedron vertex is substantially136.

l 1. A structure as defined in claim 2, wherein n=620 and thearrangement of the structural elements is obtained by triangulation fromthat structure wherein n=60.

12. A structure as defined in claim 2, wherein all of the interiorangles of all of the irregular hexagons lie in the range 114 to 126.

13. A segment of a self-supporting sphere-approximating structurecomprising, in its complete sphere-approximating version, interconnectedstructural elements forming twelve equal regular pentagons having theircenters located at the vertexes of a regular icosahedron, and, for asphere approximation of order 1, 6O hexagons; and for a sphereapproximation of order k, M, hexagons at least some of which depart fromregular hexagons to enable contiguity of adjacent ones of saidstructural elements, where k is an integer greater than l, where M,equals 3N 20, and where N is the number of hexagons for a sphereapproximation of order one less than k.

14. The structure defined in claim 13 wherein the segment approximates adome shape.

15. A segment of a self-supporting sphere-approximating structurecomprising in its whole version, interconnected structural elementsforming twelve equal regular pentagons and n hexagons at least some ofwhich hexagons are irregular to enable contiguity of adjacent modules,the centers of the pentagons defining vertexes of a regular icosahedron,where n is an integer selected from the following group: 60, 200, and620.

16. A structure as defined in claim 15, wherein n=620 and thearrangement of the structural elements is obtained by triangulation fromthat structure wherein n=60.

17. A structure as defined in claim 15, wherein all of the interiorangles of all of the irregular hexagons lie in the range 1 14 to 126.

18. The structure defined in claim 17 wherein the segment approximates adome shape.

19. A structure as defined in claim 18 wherein the structural elementsare planar pentagonal and hexagonal modules.

20. A structure as defined in claim 18, wherein the structural elementsare ribs which lie along the edges of the hexagons and pentagons.

21. A structure as defined in claim 18, wherein n=60 and all of thehexagons are of the same size and the same shape.

22. A structure as defined in claim 21 wherein a circle may be inscribedin each hexagon and in each pentagon, and the structural elements arealigned such that circles inscribed in the hexagons and pentagons wouldbe tangent to a common sphere.

23. A structure as defined in claim 18, wherein n=200and hexagons areregular and of equal size, while 60 hexagons are of a firstpredetermined size and shape, and 60 hexagons are of a secondpredetermined size and shape.

24. A structure as defined in claim 23 wherein the structural elementsare aligned such that closed curved figures, selected from the class offigures comprlsmg circles and approximate circles, inscribed in thehexagons and pentagons, would be approximately tangent to a commonsphere.

25. A structure as defined in claim 23 wherein the arrangement of thestructural elements is obtained substantially by triangulation of thatstructure obtained where the following conditions are satisfied: n isselected to be 60 and all of the hexagons are of the same size and sameshape.

26. A structure as defined in claim 18 wherein the angle between theline joining the center of any of said pentagons to a vertex of theinvert dodecahedron corresponding to the said icosahedron and the linejoining the center of the last mentioned pentagon to the vertex of thelast mentioned pentagon nearest said invert dodecahedron vertex issubstantially 136.

1. A self-supporting sphere-approximating structure comprising,interconnected structural elements forming twelve equal regularpentagons having their centers located at the vertexes of a regularicosahedron, and, for a sphere approximation of order 1, 60 hexagons;and for a sphere approximation of order k, Nk hexagons at least some ofwhich depart from regular hexagons to enable contiguity of adjacent onesof said structural elements, where k is an integer greater than 1, whereNk equals 3Nk 1 + 20, and where Nk 1 is the number of hexagons for asphere approximation of order one less than k.
 2. A self-supportingsphere-approximating structure, comprising interconnected structuralelements forming twelve equal regular pentagons and n hexagons at leastsome of which hexagons are irregular to enable contiguity of adjacentmodules, the centers of the pentagons defining vertexes of a regularicosahedron, where n is an integer selected from the following group:60, 200, and
 620. 3. A structure as defined in claim 2 wherein thestructural elements are planar pentagonal and hexagonal modules.
 4. Astructure as defined in claim 2 wherein the structural elements are ribswhich lie along the edges of the hexagons and pentagons.
 5. A structureas defined in claim 2, wherein n 60 and all of the hexagons are of thesame size and the same shape.
 6. A structure as defined in claim 2,wherein n 200 and 80 hexagons are regular and of equal size, while 60hexagons are of a first predetermined size and shape, and 60 hexagonsare of a second predetermined size and shape.
 7. A structure as definedin claim 5 wherein a circle may be inscribed in each hexagon and in eachpentagon, and the structural elements are aligned such that circlesinscribed in the hexagons and pentagons would be tangent to a commonsphere.
 8. A structure as defined in claim 6 wherein the structuralelements are aligned such that closed curved figures, selected from theclass of figures comprising circles and approximate circles, inscribedin the hexagons and pentagons, would be approximately tangent to acommon sphere.
 9. A structure as defined in claim 6 wherein thearrangement of the structural elements is obtained substantially bytriangulation of that structure obtained where the following conditionsare satisfied: n is selected to be 60 and all of the hexagons are of thesame size and same shape.
 10. A structure as defined in claim 2 whereinthe angle between the line joining the center of any of said pentagonsto a vertex of a dodecahedron corresponding in invert relationship tosaid icosahedron and the line joining the center of the last mentionedpentagon to the vertex of the pentagon nearest said dodecahedron vertexis substantially 13*6''.
 11. A structure as defined in claim 2, whereinn 620 and the arrangement of the structural elements is obtained bytriangulation from that structure wherein n
 60. 12. A structure asdefined in claim 2, wherein all of the interior angles of all of theirregular hexagons lie in the range 114* to 126*.
 13. A segment of aself-supporting sphere-approximating structure comprising, in itscomplete sphere-approximating version, interconnected structuralelements forming twelve equal regular pentagons having their centerslocated at the vertexes of a regular icosahedron, and, for a sphereapproximation of order 1, 60 hexagons; and for a sphere approximation oforder k, Nk hexagons at least some of which depart from regular hexagonsto enable contiguity of adjacent ones of said structural elements, wherek is an integer greater than 1, where Nk equals 3Nk 1 + 20, and where Nk1 is the number of hexagons for a sphere approximation of order one lessthan k.
 14. The structure defined in claim 13 wherein the segmentapproximates a dome shape.
 15. A segment of a self-supportingsphere-approximating structure comprising in its whole version,interconnected structural elements forming twelve equal regularpentagons and n hexagons at least some of which hexagons are irregularto enable contiguity of adjacent modules, the centers of the pentagonsdefining vertexes of a regular icosahedron, where n is an integerselected from the following group: 60, 200, and
 620. 16. A structure asdefined in claim 15, wherein n 620 and the arrangement of the structuralelements is obtained by triangulation from that structure wherein n 60.17. A structure as defined in claim 15, wherein all of the interiorangles of all of the irregular hexagons lie in the range 114* to 126*.18. The structure defined in claim 17 wherein the segment approximates adome shape.
 19. A structure as defined in claim 18 wherein thestructural elements are planar pentagonal and hexagonal modules.
 20. Astructure as defined in claim 18, wherein the structural elements areribs which lie along the edges of the hexagons and pentagons.
 21. Astructure as defined in claim 18, wherein n 60 and all of the hexagonsare of the same size and the same shape.
 22. A structure as defined inclaim 21 wherein a circle may be inscribed in each hexagon and in eachpentagon, and the structural elements are aligned such that circlesinscribed in the hexagons and pentagons would be tangent to a commonsphere.
 23. A structure as defined in claim 18, wherein n 200 and 80hexagons are regular and of equal size, while 60 hexagons are of a firstpredetermined size and shape, and 60 hexagons are of a secondpredetermined size and shape.
 24. A structure as defined in claim 23wherein the structural elements are aligned such that closed curvedfigures, selected from the class of figures comprising circles andapproximate circles, inscribed in the hexagons and pentagons, would beapproximately tangent to a common sphere.
 25. A structure as defined inclaim 23 wherein the arrangement of the structural elements is obtainedsubstantially by triangulation of that structure obtained where thefollowing conditions are satisfied: n is selected to be 60 and all ofthe hexagons are of the same size and same shape.
 26. A structure asdefined in claim 18 wherein the angle between the line joining thecenter of any of said pentagons to a vertex of the invert dodecahedroncorresponding to the said icosahedron and the line joining the center ofthe last mentioned pentagon to the vertex of the last mentioned pentagonnearest said invert dodecahedron vertex is substantially 13*6''.